Wavelets Based Numerical Techniques for Solving Integral and Integro-Differential Equations

Abstract

Integral equations have emerged as one of the most effective tools in applied mathematics. We adopted different approximation techniques to solve both linear and nonlinear integral and integro-differential equations of both integer and fractional order. The main objective of this study is to classify the approached techniques and estimate their accuracy and efficiency. Also, the concept of fractional calculus has been discussed in our work. Fractional calculus has many applications in the field of science and engineering such as electromagnetics, viscoelasticity, fluid mechanics, electrochemistry, signal processing etc. It plays an important role in the integration and differentiation of functions with non-integer orders. The present dissertation deals with the development of operational matrix method and collocation method based on different types of one-dimensional and two-dimensional wavelets for solving linear and nonlinear integral and integro-differential equations. The purpose is to analyse the accuracy and efficiency of the wavelet methods of solving the proposed equations. The preliminary concept of integral equations and integro-differential equations along with different kinds of kernels and the introductory concept of fractional calculus have been described in Chapter 1. In Chapter 2, basic definitions and properties of different types of wavelets such as Legendre wavelet, Bernoulli wavelet, Jacobi wavelet, Gegenabuer wavelet, CAS wavelet, Mu¨ntz-Legendre wavelet, Euler wavelet, Taylor wavelet and Laguerre wavelet have been discussed. In Chapter 3, a novel technique based on Bernoulli wavelets has been proposed to solve two-dimensional Fredholm integral equation of second kind. Bernoulli wavelets have been created by dilation and translation of Bernoulli polynomials. Also, a collocation scheme based on Laguerre wavelets has been introduced for solving the linear and nonlinear Fredholm integral equations as well as the linear system of Fredholm integral equations with weakly singular logarithmic kernel. The approached techniques have been used to transform the respective proposed equations into an algebraic system of equations. Moreover, the obtained simulation results of the several experiments are also presented in both tabular and graphical form to describe the efficiency and applicability of the approached scheme. In Chapter 4, two numerical methods, known as Gegenbauer wavelet method and CAS wavelet method have been introduced for the numerical simulation of two-dimensional Volterra integral equation. In addition, a collocation-based method based on the Jacobi wavelets method has been studied for the solution of system of two-dimensional Volterra integral equations. Finally, a brief discussion including numerical problems are presented in order to show the accuracy and effectiveness of the proposed schemes. In Chapter 5, an effective approach has been proposed to obtain the approximate solutions of linear and nonlinear two-dimensional Volterra integro-differential equations. The operational matrices of integration, differentiation, and product based on two-dimensional Bernoulli wavelets have been constructed. By utilizing the properties and matrices of wavelets along with the collocation point, the two-dimensional linear and nonlinear Volterra integro-differential equations are reduced into the system of linear and nonlinear algebraic equations respectively. The convergence analysis and error analysis have been extensively studied by the help of two-dimensional wavelets approximation. Comparison of error values and some figures obtained by the proposed wavelets have been presented in order to justify the error analysis of the proposed method. In Chapter 6, an operational matrix scheme based on two-dimensional wavelets has been introduced for the solution of weakly singular linear partial integro-differential equations and nonlinear partial integro- differential equations. By implementing two-dimensional wavelets approximations and its operational matrices of integration and differentiation along with collocation points, the linear and nonlinear weakly singular PIDEs are reduced into the system of linear and nonlinear algebraic equations respectively. Some numerical examples are included to establish the accuracy of the proposed scheme via Bernoulli wavelet approximation and Legendre wavelet approximation respectively. In Chapter 7, an effective numerical framework has been developed to obtain the solution of the Pantograph Volterra delay-integro differential equation. By utilizing the Mu¨ntz-Legendre wavelet based operational matrices along with the collocation points, the one-dimensional Pantograph Volterra delay-integro-differential equation has been reduced into an explicit system of algebraic equations. Also, a numerical operational matrix approach based on Euler wavelet is proposed to solve the nonlinear Pantograph Volterra delay-integro-differential equation of fractional order. Additionally, some numerical problems are solved to justify the applicability and validity of the presented techniques. In Chapter 8, the main objective is to establish a fractional-order operational matrix method based on Euler wavelet for solving linear Volterra-Fredholm integro differential equations with weakly singular kernels. Again, an approximation technique based on Taylor wavelet has been employed for the solution of linear and nonlinear fractional order Volterra integro-differential equation with weakly singular kernels. The results of numerical experiments have also been reported in both graphical and tabular form to illustrate the efficiency and validity of the presented methodologies. In Chapter 9, a numerical operational matrix approach based on Taylor wavelet method is proposed for solving the linear and nonlinear Volterra-Fredholm integro-differential equations. Several theorems are presented to establish the convergence and error analysis of the proposed method. Additionally, several numerical problems are included to justify the efficiency and validity of the presented technique.